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Symmetries of Spatial Graphs and Rational Twists Along Spheres and Tori Symmetry 2012, 4, 26-38; doi:10.3390/sym4010026 OPEN ACCESS symmetry ISSN 2073-8994 www.mdpi.com/journal/symmetry Article Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori Toru Ikeda Department of Mathematics, Faculty of Science, Kochi University, 2-5-1 Akebono-cho, Kochi-Shi, Kochi 780-8520, Japan; E-Mail: [email protected] Received: 14 November 2011; in revised form: 12 January 2012 / Accepted: 13 January 2012 / Published: 20 January 2012 Abstract: A symmetry group of a spatial graph Γ in S3 is a finite group consisting of orientation-preserving self-diffeomorphisms of S3 which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on a regular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ . Keywords: 3-manifold; geometric topology; symmetry; finite group action; spatial graph; rational twist 1. Introduction There are several approaches to the theory of graphs embedded in the 3-sphere, which are often motivated by molecular chemistry, since the chemical properties of a molecule depend on the symmetries of its molecular bond graph (see, for example, [1]). The symmetries of an abstract graph Γ are described by automorphisms. If Γ is embedded in S3, some of these automorphisms are induced from self-diffeomorphisms of S3. For example, [2–6] studied the extendabilities of the automorphisms of Γ , mainly in the case of Mobius¨ ladders, complete graphs, and 3-connected graphs. Even if the automorphisms of Γ extend to self-diffeomorphisms of S3, we face the problem of the uniqueness of the extensions. In this situation, it is enough to consider Γ to be a topological space, since we need to study self-diffeomorphisms of S3 which agree on Γ . In the case of a non-torus knot in S3, there are only finitely many conjugacy classes of symmetries (see [7,8]). For a cyclic period or a free period of a knot in S3, it is shown in [9,10] that the cyclic group generated by the periodic self-diffeomorphism of S3 defining the symmetry is unique up to conjugate in some cases. Moreover, Symmetry 2012, 4 27 the author [11] generalized this result to the case of links in S3. In this paper, we generalize these results to the case of symmetries of spatial graphs in S3. Suppose that any component of Γ is a non-trivial graph with no leaf. We see Γ as a geometric simplicial complex, and denote by jΓ j the underlying topological space of Γ . A tame embedding of jΓ j into S3 is called a spatial embedding of Γ into S3, or simply a spatial graph Γ in S3. We say that Γ is splittable if there exists a sphere in S3 disjoint from Γ that separates the components of Γ . We say that Γ is non-splittable if it is not splittable. Suppose that an incompressible torus in S3 − Γ bounds a solid torus V in S3 containing Γ . The core of V is called a companion knot of Γ if it is not ambient isotopic to Γ in V . If there is no companion knot of Γ , every incompressible torus in S3 − Γ separates the components of Γ . Let M be a 3-manifold, and X a submanifold of M. Denote by N(X) a regular neighborhood of X, and by E(X) = M − intN(X) the exterior of X. We refer to a finite subgroup G of the diffeomorphism group Diff(M) as a finite group action on M. Finite group actions G1 and G2 on M are equivalent (relative to X) if some h 2 Diff(M) conjugates G1 to G2 (and restricts to the identity map on X). A symmetry group G of a spatial graph Γ in S3 is a finite group action on the pair (S3;Γ ) which preserves the orientation of S3. 2 3 1 3 3 Let S be the unit sphere in R , and S the unit circle in the xy-plane in R . Denote by Rotθ 2 Diff(R ) 2 1 1 the rotation about the z-axis through angle θ. Suppose that σn 2 Diff(S ×I) and τn 2 Diff(S ×S ×I), where n 2 R, is given by σn(x; t) = (Rot2πnt(x); t) and τn(x; y; t) = (Rot2πnt(x); y; t). Let F be a 2-sided sphere or torus embedded in a 3-manifold M. Split M open along F into a (possibly disconnected) 3-manifold MF . Denote by F− and F+ the boundary components of MF originated from F . An n-twist along F is a discontinuous map on M induced from a diffeomorphism on MF −F− which restricts to the identity map on E(F+) and the map on N(F+) conjugate to σn or τn according as F is a sphere or not. We say that the n-twist is rational if n 2 Q. Figure 1 illustrates a rotational symmetry of S3 with a setwise invariant sphere S, and its conjugate by a 1=2-twist along S. Figure 1. Conjugation by a 1=2-twist along a sphere S. Our main theorem is the following: 3 Theorem 1.1. Let Γ be a spatial graph in S with no companion knot. Suppose that G1 and G2 are symmetry groups of Γ such that (1) G1(γ) = G2(γ) = γ for at least one component γ of Γ , (2) either Γ is non-splittable, or G1 and G2 are cyclic groups acting on Γ freely, and (3) G1 and G2 agree on N(Γ ). Then there is a finite sequence of rational twists along incompressible spheres and tori in E(Γ ) whose composition conjugates G2 to a symmetry group of Γ equivalent to G1 relative to N(Γ ). Symmetry 2012, 4 28 This paper is arranged as follows. In Section 2, we study symmetry groups of non-splittable spatial graph in terms of the equivariant JSJ decomposition of the exteriors. In Section 3, we establish a canonical version of the equivariant sphere theorem for the exteriors of spatial graphs with cyclic symmetry groups, and prove Theorem 1.1. 2. Non-splittable Case For a non-splittable spatial graph Γ in S3 with a non-trivial symmetry group, there is a canonical method for splitting E(Γ ) equivariantly into geometric pieces by the loop theorem, the Dehn’s lemma, and the JSJ decomposition theorem (see [12–14]). Let M be a Haken 3-manifold with incompressible boundary. The JSJ decomposition theorem and Thurston’s uniformization theorem [15] assert that there is a canonical way of splitting the pair (M; @M) along a disjoint, non-parallel, essential annuli and tori into pieces (Mi;Fi) each of which is one of the following four types: (1) Mi is an I-bundle over a compact surface and Fi is the @I-subbundle, (2) Mi admits a Seifert fibration in which Fi is fibered, (3) intMi admits a complete hyperbolic structure of finite volume, and (4) the double of (Mi;Fi − intΦi) along a non-empty compact submanifold Φi of Fi is of type (3). For a finite group action G on M, the fixed point set Fix(G) of G is the set of points in M each of which has the stabilizer G. The singular set Sing(G) of G is the set of points in M each of which has a non-trivial stabilizer. 3 Lemma 2.1. Let T be a torus embedded in S . Suppose that G1 and G2 are orientation-preserving finite group actions on S3 such that (1) G1(N(T )) = G2(N(T )) = N(T ), (2) G1 and G2 do not interchange the components of @N(T ), and (3) G1 and G2 agree on @N(T ). b 3 Then a rational twist along a component of @N(T ) conjugates G2 to a finite group action G2 on S such b that the actions of G1 and G2 on N(T ) are equivalent relative to @N(T ). Proof. It is enough by Lemma 2.4 of [11] to consider the case where the actions of G1 and G2 on N(T ) ∼ are not free. For each Gi, Theorem 2.1 of [16] implies that N(T ) = T × I admits a Gi-invariant product structure Pi, in which Sing(Gi) \ N(T ) consists of I-fibers. Since each element of Gi takes a meridian of T to a meridian of T , the setwise stabilizer of each I-fiber is a trivial group or a 2-fold cyclic group. Therefore, the quotient space N(T )=Gi admits the induced I-bundle structure over a 2-orbifold B with underlying surface F and n cone points of index two. Since T is a torus, the orbifold Euler characteristic χorb(B) of B is calculated as follows (see [17]): χorb(B) = χ(F ) − n=2 = 0: Symmetry 2012, 4 29 Since n > 0, F is a sphere and n = 4 holds. Denote by pi : N(T ) ! N(T )=Gi the projection map onto the quotient space for each i, and by Tt the T -fiber T × ftg in P1. Connect the four cone points on p1(T0) cyclically by a collection of arcs a¯1, a¯2, a¯3, and a¯4 with disjoint interiors. Each a¯i lifts to an essential loop ai on T0 such that ai and aj with i =6 j are disjoint if jj − ij = 2, and otherwise ai meets aj transversally in a point.
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